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Theorem maduval 19430
Description: Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
Hypotheses
Ref Expression
madufval.a  |-  A  =  ( N Mat  R )
madufval.d  |-  D  =  ( N maDet  R )
madufval.j  |-  J  =  ( N maAdju  R )
madufval.b  |-  B  =  ( Base `  A
)
madufval.o  |-  .1.  =  ( 1r `  R )
madufval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
maduval  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Distinct variable groups:    i, N, j, k, l    R, i, j, k, l    i, M, j, k, l
Allowed substitution hints:    A( i, j, k, l)    B( i, j, k, l)    D( i, j, k, l)    .1. ( i, j, k, l)    J( i, j, k, l)    .0. ( i, j, k, l)

Proof of Theorem maduval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . . 5  |-  A  =  ( N Mat  R )
2 madufval.b . . . . 5  |-  B  =  ( Base `  A
)
31, 2matrcl 19204 . . . 4  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 457 . . 3  |-  ( M  e.  B  ->  N  e.  Fin )
5 mpt2exga 6859 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )
64, 4, 5syl2anc 659 . 2  |-  ( M  e.  B  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e. 
_V )
7 oveq 6283 . . . . . . . 8  |-  ( m  =  M  ->  (
k m l )  =  ( k M l ) )
87ifeq2d 3903 . . . . . . 7  |-  ( m  =  M  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
98mpt2eq3dv 6343 . . . . . 6  |-  ( m  =  M  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
1093ad2ant1 1018 . . . . 5  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
1110fveq2d 5852 . . . 4  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
1211mpt2eq3dva 6341 . . 3  |-  ( m  =  M  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
13 madufval.d . . . 4  |-  D  =  ( N maDet  R )
14 madufval.j . . . 4  |-  J  =  ( N maAdju  R )
15 madufval.o . . . 4  |-  .1.  =  ( 1r `  R )
16 madufval.z . . . 4  |-  .0.  =  ( 0g `  R )
171, 13, 14, 2, 15, 16madufval 19429 . . 3  |-  J  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
1812, 17fvmptg 5929 . 2  |-  ( ( M  e.  B  /\  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )  -> 
( J `  M
)  =  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
196, 18mpdan 666 1  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058   ifcif 3884   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7553   Basecbs 14839   0gc0g 15052   1rcur 17471   Mat cmat 19199   maDet cmdat 19376   maAdju cmadu 19424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-slot 14843  df-base 14844  df-mat 19200  df-madu 19426
This theorem is referenced by:  maducoeval  19431
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